A simulated annealing-based permutation method and experimental analysis for multiple criteria decision analysis with interval type-2 fuzzy sets

Applied Soft Computing 36 (2015) 57–69

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A simulated annealing-based permutation method and experimental analysis for multiple criteria decision analysis with interval type-2 fuzzy sets Jih-Chang Wang a,1 , Ting-Yu Chen b,∗ a

Department of Information Management, College of Management, Chang Gung University, 259, Wen-Hwa 1st Road, Kwei-Shan, Taoyuan 333, Taiwan Department of Industrial and Business Management, Graduate Institute of Business and Management, College of Management, Chang Gung University, 259, Wen-Hwa 1st Road, Kwei-Shan, Taoyuan 333, Taiwan b

a r t i c l e

i n f o

Article history: Received 14 October 2014 Received in revised form 27 June 2015 Accepted 15 July 2015 Available online 26 July 2015 Keywords: Simulated annealing Permutation method Multiple criteria decision analysis Interval type-2 fuzzy set Incomplete preference information Computational experiment

a b s t r a c t The aim of this paper is to develop a simulated annealing-based permutation method for multiple criteria decision analysis within the environment of interval type-2 fuzzy sets. The outranking methodology constitutes one of the most fruitful approaches in multiple criteria decision making and has been applied in numerous real-world problems. The permutation method is a classical outranking model, which generalizes Jacquet–Lagreze’s permutation method and is based on a pairwise criterion comparison of the alternatives. Because modeling of the uncertainty in the decision-making process becomes increasingly important, an extension to the interval type-2 fuzzy environment is a useful generalization of the permutation method and is appropriate for handling uncertain and imprecise information in practical decision-making situations. This paper produces a signed-distance-based comparison among the comprehensive rankings of alternatives for concordance and discordance analyses. An integrated nonlinear programming model is constructed for estimation of the criterion weights and the optimal ranking order of the alternatives under incomplete preference information. To enhance the implementation efficiency, a simulated annealing-based permutation method and its meta-heuristic algorithm are developed to produce a polynomial time solution for the total completion time problem. Furthermore, computational experiments with notably large amounts of simulation data are conducted to test the solution approach and validate the correctness of the approximate solution compared with the optimal all-permutation-based result. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Multiple criteria decision analysis (MCDA) problems address the ranking of alternatives and the selection of the best alternative among a finite set of alternatives based on a finite set of criteria with incomplete information [1,2]. The outranking methodology constitutes one of the most fruitful approaches in MCDA and has been applied in many real-life problems [3]. Specifically, the outranking model arranges a set of preference rankings that best satisfy a given concordance measure [4]. In particular, the permutation method from Paelinck’s qualitative multiple criteria analysis [5,6] is

∗ Corresponding author. Tel.: +886 3 2118800x5678; fax: +886 3 2118500. E-mail addresses: [email protected] (J.-C. Wang), [email protected] (T.-Y. Chen). 1 Tel.: +886 3 2118800x5821; fax: +886 3 2118700. http://dx.doi.org/10.1016/j.asoc.2015.07.011 1568-4946/© 2015 Elsevier B.V. All rights reserved.

a classical outranking model in MCDA. This method, which generalizes Jacquet–Lagreze’s permutation method, is a metric procedure and is based on an evaluation of all possible rankings (permutations) of the alternatives under consideration [3]. Uncertain and imprecise assessment of information often occurs in practical decision-making situations [7,8]. Thus, modeling of uncertainty in subjective judgments and evaluation processes becomes increasingly important in handling MCDA problems [9,10]. Accordingly, an extension to the fuzzy environment is a natural generalization of the permutation-based outranking methodology. For example, Chen et al. [11] developed an intuitionistic fuzzy permutation method for addressing MCDA problems. Chen and Wang [12] proposed an interval-valued fuzzy permutation method and conducted an experimental analysis using cardinal and ordinal evaluations. Nevertheless, available information is sometimes not sufficient for the exact definition of a degree of membership for certain elements [7]. It is not reasonable to use an accurate membership

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function for a factor that is uncertain [13–15]. Therefore, type-1 fuzzy sets (T1 FSs) cannot fully address all of the uncertainty present in real-world problems [16]. The concept of type-2 fuzzy sets (T2 FSs) is an extension of T1 FSs [17,18]. T2 FSs are superior to T1 FSs because T2 FSs can model second-order uncertainties [19,20]. Unfortunately, the computational complexity of using T2 FSs is very high, which makes it very difficult to employ them in practical applications [21,22] (Zhang, 2013). Due to the high computational complexity of using T2 FSs, interval type-2 fuzzy sets (IT2 FSs) have become the most widely used T2 FSs [23–28]. This paper attempts to focus on the permutation-based outranking methodology within the decision environment of IT2 FSs. In addition to considering the context of IT2 FSs, Chen et al. [29] developed an extended qualitative flexible multiple criteria method (QUALIFLEX) for handling medical decision-making problems. Wang et al. [30] introduced a likelihood-based QUALIFLEX method for addressing MCDA problems. Both the extended QUALIFLEX method and the likelihood-based QUALIFLEX method belong to the permutation-based outranking methodology based on IT2 FSs. Although the feasibility and effectiveness of the above-mentioned methods in the MCDA field have been thoroughly investigated and demonstrated, the main drawback of these methods is low computation efficiency in cases of numerous alternatives. In fact, the existing permutation-based outranking methods have an evident limitation because the solution for all of the possible permutations is an NP-hard problem. The permutation methods need to list all of the possible permutations of the alternatives. If the number of alternatives is not very large, for example, seven and ten alternatives, it would need to list 5040 and 3,628,800 permutations, respectively. It seems that a simple problem would be a complicated problem when the permutation methods are used. Furthermore, this paper has conducted a pilot study to investigate the average CPU time for generating all possible permutations of the alternatives. For each number of alternatives, ten trials were implemented using the MATLAB run on an x64-based PC with an Intel Core i7-4500U (1.8 GHz) CPU, 8 G RAM, and an operating system Windows 7. The average CPU times for producing all permutations are 0.000953, 0.003609, 0.018750, 0.109375, 0.76875, 6.16875, 55.5, 556.23 (i.e., 9.27 min), and 6100 (i.e., 1.69 h) s for cases with three to eleven alternatives, respectively. Obviously, the average CPU times in the cases of ten and eleven alternatives are significantly high. In general, the number of permutations increases rapidly with an increasing number of alternatives. It directly follows that the computational complexity of the existing permutation-based outranking methods is evidently high in the case of many alternatives. Therefore, current methods will encounter great difficulties in a large-size decision-making problem. Moreover, this problem will be more serious within the IT2 FS environment because of sophisticated operations based on type-2 fuzzy logic. For the above reasons, this paper intends to develop a new permutation-based method in the context of IT2 FSs to overcome the difficulty of huge computations and to capture imprecise and uncertain information. Simulated annealing (SA) has been widely applied in NP-hard problems [31]. The SA is a meta-heuristic optimization method used to solve complex problems with large solution spaces and produces results close to the global optimum value in a short period of time [32]. This algorithm is well suited to solving large-scale and difficult optimization problems [33,34]. The concept of SA is inspired by nature and is derived from statistical mechanics [32]. The SA method emulates the solid annealing process, which first heats a solid to its melting point and subsequently slowly cools the material [35]. Specifically, the SA method is based on an analogy with thermodynamics and the manner in which liquids freeze and crystallize; moreover, the freezing

process is guided by a cooling schedule that controls the decay of the system temperature [34]. The SA method has the advantage in that it avoids becoming trapped in local optima [36]. Thus, the SA method has become one of the most popular metaheuristic methods and has been applied widely to solve many combinatorial optimization problems [35–37]. SA provides a suitable approach to solving the optimization problem in a wide range of applications [38,39] and is a stochastic optimization technique that converges on the global optimal solution. Therefore, this paper attempts to develop an SA-based permutation method to improve the computation efficiency and acquire a polynomial time solution. The purpose of this paper is to construct a new outranking approach, i.e., an SA-based permutation method, for solving MCDA problems under incomplete preference information within the environment of IT2 FSs. In the context of interval type-2 trapezoidal fuzzy numbers (IT2 TrFNs), this paper proposes the main structure of the permutation-based outranking method for addressing the problems caused by greater imprecision or uncertainty in MCDA. Instead of relying on a complicated computational process to handle IT2 TrFN data, this paper develops a simple and effective comparative method based on the concept of signed distances to differentiate the sets of concordance and discordance. The proposed method offers a flexible approach capable of tackling decision-making problems featuring conflicting information with respect to criterion importance under the incomplete preference structure. Furthermore, this paper provides an SA-based permutation algorithm for the solution approach to enhance the implementation efficiency of the proposed method, especially in case of many alternatives, and determines a polynomial time solution for the total completion time problem. This paper makes several significant contributions to the existing literature on the permutation-based outranking methodology in the MCDA field. First, we establish the fundamental structure of the SA-based permutation method based on IT2 TrFNs to address second-order uncertainties in decision reality. Second, compared to other existing permutation methods, the proposed method provides a flexible approach capable of manipulating incompletely known or even conflicting information about criterion importance in practice. Third, the proposed method uses a hybrid approach that integrates SA into the permutation-based outranking methodology to adapt to the MCDA problems within the interval type-2 fuzzy environment. Fourth, the difficulty of implementing computations for MCDA problems with numerous alternatives can be significantly overcome with the help of the proposed SA-based permutation algorithm. Fifth, the applicable scope of the problem size can be evidently expanded because the SA-based permutation algorithm can improve the computation efficiency and acquire a polynomial time solution. In contrast, the existing permutation methods can only be applied to MCDA problems with a limited number of alternatives. Finally, the computational experiments and the comparative analysis validate the effectiveness and efficiency of the proposed method for applications. Thus, the proposed SA-based permutation method not only improves the established methods but also enriches the development of various permutation-based outranking methods, especially within the decision environment of IT2 FSs. This paper is organized as follows. Section 2 briefly reviews the basic concepts of T2 FSs, IT2 FSs, and IT2 TrFNs. Section 3 formulates an MCDA problem in the IT2 TrFN framework and develops the SA-based permutation method under incomplete preference information. Section 4 conducts a comparative analysis via computational experiments to examine the effectiveness and efficiency of the proposed method. This section also consists of comparisons with other relevant methods and discusses the advantages of the proposed method. Finally, Section 5 presents the conclusions.

J.-C. Wang, T.-Y. Chen / Applied Soft Computing 36 (2015) 57–69

59

2. Basic concepts of T2 FSs and IT2 FSs Several of the relevant definitions and operations of T2 FSs, IT2 FSs, and IT2 TrFNs are briefly reviewed in this section. The membership grade of a T1 FS is a real number in the interval [0,1], whereas the membership grade of a T2 FS is a T1 FS with a support bounded by the interval [0,1] [18]. The membership functions of T1 FSs are two-dimensional and completely crisp, whereas the membership functions of T2 FSs are three-dimensional and fuzzy. The new third dimension of T2 FSs provides an additional degree of freedom and enables T2 FSs to directly model uncertainties [40–43]. Furthermore, the amount of uncertainty in MCDA problems can be reduced by using T2 FSs because they have the capability to tackle the problem of uncertain and imprecise information [44]. Let X be a crisp set. A mapping A: X → [0,1][0,1] is called a T2 FS defined on the universe of discourse X; it is represented by [45–47]:

A=





(x, A (x)) x ∈ X, A (x) =

fx (u) ∈ [0, 1]









=

,

(1)



(x, A (x)) x ∈ X, A (x)



only if ∀x ∈ X, AL (x) ≤ AU (x)). Let  ∈ {L, U}. The membership function of x in A for each  is expressed as follows [30,59]:

(u, fx (u)) u ∈ Jx ⊆ [0, 1],

where x is called a primary variable, and A (x) denotes the fuzzy membership value of x in A. Note that A (x) is also known as a secondary membership function or a secondary set. Jx ⊆ [0, 1] denotes the domain of fx (u) and represents the primary membership values of x ∈ X. Moreover, fx (u) denotes the secondary membership (grade), where u indicates the primary membership (grade) of x. IT2 FSs have a greater capacity to address linguistic uncertainties by modeling the vagueness and unreliability of information [13,14,48,49]. The computations associated with IT2 FSs are manageable [29,50], which enriches the development of various MCDA methods within the decision environment of IT2 FSs [21,27,51–56]). Next, we introduce some basic concepts of IT2 FSs. Let A be a T2 FS on X. When fx (u) = 1 for all u ∈ Jx , A is known as an IT2 FS on X and can be represented by [45,47]: A=

˜ Fig. 1. A geometrical interpretation of an IT2 TrFN A.



(u, 1) AL (x) ≤ u ≤ AU (x), [AL (x), AU (x)] ⊆ [0, 1]



, (2)

where A (x) is referred to as an interval membership value. Let A be an IT2 FS on X. A can be fully characterized by its footprint of uncertainty (FOU), which is defined as the union of all primary memberships as follows [40,45,47]:

A (x) =

⎧   ⎪ hA (x − a1 )   ⎪ ⎪ if a1 ≤ x ≤ a2 , ⎪   ⎪ a − a ⎪ 2 1 ⎪ ⎪   ⎨  hA

if a2 ≤ x ≤ a3 ,

0

otherwise.

(6)

  ⎪ ⎪ hA (a4 − x)   ⎪ ⎪ if a3 ≤ x ≤ a4 , ⎪ ⎪ ⎪ a4 − a3 ⎪ ⎩

The LMF AL (x) and the UMF AU (x) are lower and upper bounds, respectively, for the FOU(A) of A. Then, A is an IT2 TrFN on X (see Fig. 1 for a geometrical interpretation) and is represented by: U U U U A = [AL , AU ] = [(aL1 , aL2 , aL3 , aL4 ; hLA ), (aU 1 , a2 , a3 , a4 ; hA )].

(7)

3. SA-based permutation method based on IT2 TrFNs This section first formulates an MCDA problem based on IT2 TrFNs in which the information on the criterion weights is incompletely known and the criterion values take the form of IT2 TrFNs. Applying a signed-distance-based approach, this section determines the sets of concordance and discordance for each possible ranking of the alternatives. This section develops the SA-based permutation method based on IT2 TrFNs under incomplete preference information used to solve the optimal criterion weights and the priority orders of the alternatives. 3.1. MCDA problem under incomplete information

FOU(A) = ∪ [AL (x), AU (x)].

(3)

x∈X

FOU(A) is a bounded region that represents the uncertainty associated with the membership grades of A. A lower membership function (LMF) and an upper membership function (UMF) are two type-1 membership functions that are bounds for the FOU(A) of an IT2 FS A. Let two T1 FSs AL : X → [0,1] and AU : X → [0,1] be the lower and upper fuzzy sets, respectively, with respect to A. The LMF AL (x) and the UMF AU (x) are associated with the lower bound FOUL (A) and the upper bound FOUU (A), respectively, of FOU(A). They are defined as follows [22,45,57,58]: AL = FOU L (A) =





(x, AL (x)) x ∈ X



,

  AU = FOU U (A) = (x, AU (x)) x ∈ X , 

Consider an MCDA problem based on IT2 TrFNs under incomplete preference information. Define Z = {z1 , z2 , . . ., zm } as the set of decision alternatives, where m is the number of alternatives. Define C = {c1 , c2 , . . ., cn } as the set of evaluative criteria, where n is the number of criteria. The set C can be generally divided into two sets, CI and CII , where CI denotes a collection of benefit criteria (i.e., larger values of cj indicate a greater preference), CII denotes a collection of cost criteria (i.e., smaller values of cj indicate a greater preference), and where CI ∩ CII = ∅ and CI ∪ CII = C. Let wj be the weight of a criterion cj ∈ C, which satisfies the normalization conditions. Let  0 denote a set of all weight vectors, and

(4) 0 = (5)

where 0 ≤ AL (x) ≤ AU (x) ≤ 1 for all x ∈ X. Let A be an IT2 FS on X. Let AL (= (aL1 , aL2 , aL3 , aL4 ; hLA )) and AU (= , aU , aU , aU ; hU )) be the lower and upper trapezoidal fuzzy num(aU 1 2 3 4 A ≤ bers, respectively, with respect to A, where aL1 ≤ aL2 ≤ aL3 ≤ aL4 , aU 1 U ≤ aU , 0 ≤ hL ≤ hU ≤ 1, aU ≤ aL , aL ≤ aU , and AL ⊆ AU (if and aU ≤ a 2 3 4 1 4 1 4 A A



 

(w1 , w2 , . . ., wn ) wj ≥ 0(j = 1, 2, . . ., n),

n

j=1

wj = 1



. (8)

Incomplete preference information is realistic in many practical decision-making problems [60]. The incomplete information about criterion weights can be generally constructed using the following basic ranking forms [61–65].

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J.-C. Wang, T.-Y. Chen / Applied Soft Computing 36 (2015) 57–69

3.2. Proposed method

(i) A weak ranking: 1 =



 (w1 , w2 , . . ., wn ) ∈ 0  wj1 ≥ wj2 

for allj1 ∈ 1 and j2 ∈ 1 ,

(9)

Pl = (. . ., z , . . ., zϕ , . . .),

(ii) A strict ranking: 2 =



 (w1 , w2 , . . ., wn ) ∈ 0  wj1 − wj2 ≥ ıj1 j2 

for all j1 ∈ 2 and j2 ∈ 2 ,

(10)

(iii) A ranking of differences (or strength of preference): 3 =





(w1 , w2 , . . ., wn ) ∈ 0  wj1 − wj2 ≥ wj2 − wj3



for all j1 ∈ 3 , j2 ∈ 3 , and j3 ∈ ˝3 ,

(11)

4 =

(12)



1 2

for all j1 ∈ 5 and j2 ∈ 5 .

(13)

In (9)–(13),  1 ,  2 , . . .,  5 , 1 , 2 , 3 , 5 , and ˝3 are denoted as the subsets of the subscript index set N = {1, 2, . . ., n} of all criteria, where  1 and 1 , as well as  2 and 2 ,  3 , 3 , and ˝3 , and  5 and 5 , are disjoint. Moreover, ıj1 j2 , ı j , εj1 , and ı j j are constants that 1

1

1

1

1 2

Let  denote a set of the incompletely known information about the criterion weights provided by the decision maker. Because the preference information regarding the criterion importance may consist of several sets of the five basic ranking forms,  consists of the above five sets  1 ,  2 , . . ., and  5 . That is,  = 1 ∪ 2 ∪ 3 ∪ 4 ∪ 5 .



Aij = [ALij , AU ] = (aL1ij , aL2ij , aL3ij , aL4ij ; hLA ), (aU , aU , aU , aU ; hU A ) , ij 1ij 2ij 3ij 4ij ij

where 0 ≤ aL1ij ≤ aL2ij ≤ aL3ij ≤ aL4ij , 0 ≤ aU ≤ aU ≤ aU ≤ aU , aU ≤ 1ij 2ij 3ij 4ij 1ij aL1ij , aL4ij ≤ aU , 0 ≤ hLA ≤ hU ≤ 1, and ALij ⊆ AU . A 4ij ij ij

+

+ 3hLA 2hU A j

aL1j + aL2j + aL3j + aL4j + 4hU − 3hLA A j

j

hU A

+ aU ) 4j

(aU + aU ) 2j 3j

j

 (aU 1j

j

hU A ,

(17)

j

 1 sd(Aϕj ) = 8

+

aL1ϕj + aL2ϕj + aL3ϕj + aL4ϕj + 4hU − 3hLA A

ϕj

ϕj

hU A

+ 3hLA 2hU A ϕj

+ aU ) 4ϕj

(aU + aU ) 2ϕj 3ϕj

ϕj

 (aU 1ϕj

ϕj

hU A .

(18)

ϕj

Let us take Aj = [(0.72, 0.74, 0.82, 0.85 ; 0.80), (0.65, 0.70, 0.86, 0.92 ; 1.00)] as an example. Applying (17), the signed distance sd(Aj ) is obtained as follows: sd(Aj ) =



1 2 × 1.0 + 3 × 0.8 0.72 + 0.74 + 0.82 + 0.85 + 8 1.0 ×(0.70 + 0.86) +

4 × 1.0 − 3 × 0.8 (0.65 + 0.92) =1.56. 1.0

ij

(15)

ij

 1 sd(Aj ) = 8

(14)

Let us consider an example. Assume that C = {c1 , c2 , c3 } and the incomplete information about criterion weights provided by the decision maker is indicated as follows: w2 ≥ w1 , w2 − w3 ≥ 0.2, w2 − w1 ≥ w1 − w3 , 0.1 ≤ w1 ≤ 0.3, and 0.1 ≤ w1 ≤ 0.3. Applying (9)-(13), it follows that 1 = {w2 ≥ w1 }, 2 = {w2 − w3 ≥ 0.2} (ı23 = 0.2), 3 = {w2 − w1 ≥ w1 − w3 }, 4 = {0.1 ≤ w1 ≤ 0.3} (ı 1 = 0.1 and ε1 = 0.2), and 5 = {w3 ≥ 0.6 · w1 } (ı 31 = 0.6). According to (14), the set  is obtained as follows:  = {w2 ≥ w1 , w2 − w3 ≥ 0.2, w2 − w1 ≥ w1 − w3 , 0.1 ≤ w1 ≤ 0.3, w3 ≥ 0.6 · w1 }. Decision makers often use linguistic variables to evaluate the ratings of alternatives with respect to various criteria [28,66]. These linguistic values can be represented with IT2 TrFNs in the IT2 FS context [26,27,29,60]. Let a non-negative IT2 TrFN Aij denote the evaluative rating of an alternative zi ∈ Z with respect to a criterion cj ∈ C. Aij is represented as follows:

ϕj

Note that Aj and Aϕj are IT2 TrFNs and cannot be easily compared. A signed distance (i.e., an oriented distance or directed distance) has often been employed to determine the rankings of fuzzy numbers [20]. The concept of signed distances has been extended to the context of IT2 TrFNs; moreover, the signed-distance-based approach has been successfully applied to develop several useful MCDA methods [20,21,26,29,54,60,66]. Accordingly, this paper uses a signed-distance-based approach presented by Chen [60]; Chen [20]; Chen [26] and Chen et al. [29] to obtain comparable values of Aj and Aϕj for , ϕ = 1, 2, . . ., m and  = / ϕ. Let sd(Aj ) and sd(Aϕj ) denote the signed distances from Aj and Aϕj , respectively, to the level-1 fuzzy number mapping on the vertical axis at the origin of the coordinates. sd(Aj ) and sd(Aϕj ) are calculated as follows:

1 2

satisfy the conditions ıj1 j2 > 0, ı j ≥ 0, εj1 ≥ 0, 0 ≤ ı j ≤ ı j + εj1 ≤ 1, and 0 ≤ ı j j ≤ 1.

j

ϕj

(w1 , w2 , . . ., wn ) ∈ 0  wj1 ≥ ı j j · wj2



where the alternative z is ranked higher than or equal to zϕ . That is, Pl represents the following hypothesis: · · · - z - · · · - zϕ - · · ·. Assume that Z = {z1 , z2 , z3 } for example. There are 6 (=3!) permutations of the ranking for the three alternatives (z1 , z2 , and z3 ) that have to be evaluated, including P1 = (z1 , z2 , z3 ), P2 = (z1 , z3 , z2 ), P3 = (z2 , z1 , z3 ), P4 = (z2 , z3 , z1 ), P5 = (z3 , z1 , z2 ), and P6 = (z3 , z2 , z1 ). Next, this paper employs a signed-distance-based approach to identify the sets of concordance and discordance. Consider the two evaluative ratings Aj and Aϕj , where Aj = [ALj , AU ]= j ] = [(aL1ϕj , aL2ϕj , aL3ϕj , aL4ϕj ; hLA ), (aU , aU , aU , aU ; hU )]. [ALϕj , AU A ϕj 1ϕj 2ϕj 3ϕj 4ϕj

(v) A ratio bound (or a ranking with multiples):



(16)

j

 (w1 , w2 , . . ., wn ) ∈ 0  ı j + εj1 ≥ wj1 ≥ ı j 1 1 

for all j1 ∈ 4 ,

5 =

for l = 1, 2, . . ., m!,

[(aL1j , aL2j , aL3j , aL4j ; hLA ), (aU , aU , aU , aU ; hU )] and Aϕj = A 1j 2j 3j 4j

(iv) An interval bound:



In the addressed MCDA problem, m! permutations of the possible rankings of the alternatives exist. Let Pl denote the lth permutation as follows:

The measurement definitions in (17) and (18) have been proven to satisfy several important properties. For example, Chen [60] has proven that the signed distance based on IT2 TrFNs satisfies the law of trichotomy. Chen [54] has investigated the appropriateness

J.-C. Wang, T.-Y. Chen / Applied Soft Computing 36 (2015) 57–69

of the linear order via the signed-distance-based approach. In this paper, the main reason for employing signed distances is the fact that it is simple and effective to compare IT2 TrFN data within the environment of IT2 FSs. The signed-distance-based approach can simplify the ranking procedure of IT2 TrFNs. More importantly, the signed-distance-based approach can apply both positive and negative values to define the ordering of IT2 TrFNs, an approach that differs considerably from that of ordinary distance measures. Because signed distances provide a linear order that allows comparing IT2 TrFNs [54], the concept of signed distances is suggested for use in acquiring comparable values of the evaluative ratings in the proposed SA-based permutation method. According to the inequality relations of the signed distances sd(Aj ) and sd(Aϕj ), the sets of concordance and discordance are defined as follows:

E(Pl ) =





sd(Aj ) − sd(Aϕj ) · wj

,ϕ cj ∈CSϕ ∩CI



−





sd(Aϕj ) − sd(Aj ) · wj

,ϕ cj ∈DSϕ ∩CI



+





sd(Aϕj ) − sd(Aj ) · wj

,ϕ cj ∈CSϕ ∩CII



−







sd(Aj ) − sd(Aϕj ) · wj

,ϕ cj ∈DSϕ ∩CII

=



⎡



+

(19)



⎣





sd(Aj ) − sd(Aϕj ) · wj

cj ∈(CSϕ ∪DSϕ )∩CI

,ϕ

    CSϕ = cj  sd(Aj ) > sd(Aϕj ) cj ∈ CI ,    sd(Aj ) < sd(Aϕj ) cj ∈ CII ,



61



⎤



sd(Aϕj ) − sd(Aj ) · wj ⎦

cj ∈(CSϕ ∪DSϕ )∩CII

=





  sd(Aj ) − sd(Aϕj ) · wj ,

(21)

,ϕ cj ∈CSϕ ∪DSϕ

    cj  sd(Aj ) < sd(Aϕj ) cj ∈ CI ,    sd(Aj ) > sd(Aϕj ) cj ∈ CII ,

DSϕ =

(20)

where (. . . , , . . ., ϕ, . . .) is a permutation of (1, 2, . . ., m), , ϕ = 1, 2, . . ., m, and  = / ϕ. The concordance set CSϕ is the subset of all criteria for which sd(Aj ) > sd(Aϕj ) for cj ∈ CI and sd(Aj ) < sd(Aϕj ) for cj ∈ CII . The discordance set DSϕ is the subset of all criteria for which sd(Aj ) < sd(Aϕj ) for cj ∈ CI and sd(Aj ) > sd(Aϕj ) for cj ∈ CII . In a particular ranking, if the partial ranking z zϕ appears, sd(Aj ) > sd(Aϕj )









for each l = 1, 2, · · ·, m!. Recall that  is a set of the information on the criterion weights that is incompletely known to the decision maker. If the set  contains inconsistent preference information, the conditions in  should be relaxed to  by introducing the non-negative deviation − − − − + − , e(ii)j , e(iii)j , e(iv)j , e(iv)j , and e(v)j , which are variables e(i)j j j j j j 1 2

defined as follows:





1 =

1

1

1 2





− (w1 , w2 , · · ·, wn ) ∈ 0  wj1 + e(i)j

1 j2



≥ wj2

for all j1 ∈ 1 and j2 ∈ 1 .



be rated − sd(Aϕj ) − sd(Aj ) · wj and sd(Aϕj ) − sd(Aj ) wj for cj ∈ CI and cj ∈ CII , respectively. For example, assume that the decision maker would like to evaluate the three alternatives in Z = {z1 , z2 , z3 } using the three criteria in C = {c1 , c2 , c3 }, where CI = {c1 , c3 } and CII = {c2 }. Let us consider z1 and z3 in a particular permutation P2 = (z1 , z3 , z2 ) (with the corresponding hypothesis z1 - z3 - z2 ) as an example. It is given that A11 = [(0.83, 0.87, 0.92, 0.94; 0.80), (0.78, 0.84, 0.95, 0.98; 1.00)], A12 = [(0.12, 0.17, 0.23, 0.25; 0.80), (0.10, 0.15, 0.29, 0.38; 1.00)], A13 =[(0.78, 0.80, 0.85, 0.88; 0.80), (0.73, 0.77, 0.88, 0.93; 1.00)], A31 = [(0.94, 0.96, 0.98, 0.98; 0.80), (0.92, 0.96, 0.99, 1.00; 1.00)], A32 = [(0.13, 0.15, 0.19, 0.22; 0.80), (0.11, 0.14, 0.21, 0.31; 1.00)], and A33 = [(0.62, 0.64, 0.72, 0.75; 0.80), (0.55, 0.60, 0.76, 0.82; 1.00)]. Applying (17) and (18), the signed distances are obtained as follows: sd(A11 ) = 1.78, sd(A12 ) = 0.43, sd(A13 ) = 1.65, sd(A31 ) = 1.94, sd(A32 ) = 0.36, and sd(A33 ) = 1.36. Observe that sd(A13 ) > sd(A33 ), so the concordance set CS13 = {c3 } can be obtained using (19). Therefore, the concordance testing result concerning z1 and z3 is 0.29 w3 (i.e., (1.65 − 1.36) · wj ). Based on (20), it is known that the discordance set DS13 = {c1 , c2 } because sd(A11 ) < sd(A31 ) and sd(A12 ) > sd(A32 ). The discordance testing result concerning z1 and z3 is 0.16 · w1 + 0.07 · w2 (i.e., −(1.78 − 1.94) · w1 + (0.43 − 0.36) · w2 ). The evaluation value of the chosen hypothesis for ranking the alternatives is the algebraic sum of all of the weighted differences among the signed distances corresponding to the criterion-by-criterion consistency. The evaluation value E(Pl ) of the lth permutation Pl is defined as follows:

1 2 3

(i’) A relaxed weak ranking:

will be rated sd(Aj ) − sd(Aϕj ) · wj and − sd(Aj ) − sd(Aϕj ) · wj for cj ∈ CI and cj ∈ CII , respectively. In contrast, sd(Aj ) < sd(Aϕj ) will



1 2

(ii’) A relaxed strict ranking: 2 =

(22)

 



− (w1 , w2 , · · ·, wn ) ∈ 0 wj1 − wj2 + e(ii)j

1 j2



≥ ıj1 j2

for all j1 ∈ 2 and j2 ∈ 2 .

(23)

(iii’) A relaxed ranking of differences: 3 =



− (w1 , w2 , · · ·, wn ) ∈ 0 |wj1 − 2wj2 + wj3 + e(iii)j

1 j2 j3



for all j1 ∈ 3 , j2 ∈ 3 , and j3 ∈ ˝3 . (iv’) A relaxed interval bound: 4 =



≥0 (24)



− (w1 , w2 , . . ., wn ) ∈ 0  wj1 + e(i v)j ≥ ıj , 1

1

+ wj1 − e(i v)j ≤ ıj + εj1 1

for all j1 ∈ 4

1



.

(25)

(v’) A relaxed ratio bound:

 5 =

  wj1

(w1 , w2 , . . ., wn ) ∈ 0 

 for all j1 ∈ 5 and j2 ∈ 5

wj2

.

+ e(−v)j

1 j2

≥ ı j j

1 2

(26)

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J.-C. Wang, T.-Y. Chen / Applied Soft Computing 36 (2015) 57–69

Additionally,

 =

1

∪ 2

∪ 3

∪ 4

∪ 5 .

(27)

Consider the aforementioned example about incomplete information again. Incorporating the non-negative deviation variables − − − − + − e(i)21 , e(ii)23 , e(iii)213 , e(iv)1 , e(iv)1 , and e(v)31 into the conditions in  1 −  5 , the relaxed sets 1 − 5 can be determined using − (22)-(26), respectively, as follows: 1 = {w2 + e(i)21 ≥ w1 }, 2 = 2

− − {w2 − w3 + e(ii)23 ≥ 0.2}, 3 = {w2 − 2w1 + w3 + e(iii)213 ≥ 0}, 4 = − + − {w1 + e(iv)1 ≥ 0.1, w1 − e(iv)1 ≤ 0.3}, and 5 = {(w3 /w1 ) + e(v)31 ≥ 0.6}. The relaxed set  can be correspondingly acquired using (27). Considering the maximal evaluation value of each permutation and minimal deviation values, the following integrated nonlinear programming model can be constructed to determine the optimal weight vector for each permutation:

t ) − E(P t ) from the original E(P t ) to measure the change in E(Pcur can cur the optimal evaluation values at the tth iteration:

E(P t ) =

t ) − E(P t ) E(Pcur can t ) E(Pcur

.

(29)

If the optimal evaluation value of the candidate permutation is larger than that of the current one, the current permutation is replaced by the candidate permutation; otherwise, a probabilistic decision for the acceptance of the worse permutation is conducted to determine whether to accept the candidate permutation. Let Tmax denote an initial temperature; moreover, let T1 = Tmax . We designate Tmax in the following: Tmax =

−0.1 . ln(0.1)

(30)

max

s.t.

⎧ E(Pl ) ≥ , ⎪ ⎪ ⎪  ⎪

 ⎪ ⎪ − − − − + − ⎪ e(i)j + e(ii)j + e(iii)j + e(iv)j + e(iv)j + e(v)j ≥ , − ⎪ j j j j j 1 2 1 2 1 2 3 1 1 1 2 ⎪ ⎪ ⎪ j ,j ,j ∈N ⎪ ⎪ 1 2 3 ⎪ (w1 , w2 , · · ·, wn ) ∈  , ⎪ ⎪ ⎪ ⎨ − e(i)j

1 j2

≥0

⎪ − ⎪ ≥0 e(ii)j ⎪ ⎪ 1 j2 ⎪ ⎪ ⎪ − ⎪ e(iii)j j j ≥ 0 ⎪ 1 2 3 ⎪ ⎪ ⎪ − + ⎪ e ≥ 0, e(i ⎪ v)j1 ≥ 0 (iv)j1 ⎪ ⎪ ⎪ ⎩ e− ≥0 (v)j1 j2

for each l = 1, 2, . . ., m!. Each of the solution results in the proposed model yields the optimal weight vector w = (w1 , w2 , . . ., wn ) for each l = 1, 2, . . ., m!. Correspondingly, the optimal evaluation value E(Pl ) of the lth permutation can be obtained. In total, m! integrated programming problems must be solved because there are m! permutations in the alternative set. One can choose the maximum value among all of the E(Pl ) values and obtain the optimal ranking order of the alternatives. However, because the solution to the proposed model in (28) for all m! permutations is an NP-hard problem, this paper establishes an SA-based algorithm for the proposed permutation outranking method to improve the computation efficiency and acquire a polynomial time solution. The SA-based permutation algorithm begins with an initial solution chosen at random from all of the m! pert be the current permutation at the tth iteration, mutations. Let Pcur where t is the iteration number, the maximal t value tmax is designated as mς (which is polynomially solvable), and ς is a positive t integer. For each iteration, a candidate permutation Pcan is generated from the neighborhood solutions of the current permutation t using a rearrangement operation. Using the exchange operator, Pcur t by choosing two alternatives at one can obtain a neighbor of Pcur random and altering their current ranking orders. In the same manner, this paper applies the exchange operator (m − i + 1) times to t , where generate a final neighbor as the candidate permutation Pcan i = 1, 2, . . ., m. Let us consider the illustrative example Z = {z1 , z2 , z3 , z4 }. Fig. 2 t . As revealed shows the approach to generating the neighbor of Pcur t in this figure, suppose that Pcur = (z2 , z3 , z1 , z4 ) and i = 1. There are four alternatives, and thus, we must implement the exchange opert ator four (i.e., 4 − 1 + 1) times to generate Pcan = (z4 , z3 , z2 , z1 ) from the neighborhood solutions. After generating the neighbor permutation, the change in the t ) and E(P t ) optimal evaluation values can be acquired. Let E(Pcur can t be the optimal evaluation values for the permutations Pcur and t , respectively. This paper uses the proportion of the difference Pcan

j1 ∈ 1

and

j2 ∈ 1 ,

j1 ∈ 2

and

j2 ∈ 2 ,

j1 ∈ 3 ,

j2 ∈ 3 ,

(28)

and j3 ∈ ˝3 ,

j1 ∈ 4 , j1 ∈ 5

and

j2 ∈ 5 ,

This paper considers m stages of the temperature change. As suggested by Maulik and Mukhopadhyay [67] and Yeh et al. [36], the current temperature Ti at the i th stage is defined as follows:

Ti = Tmax · 0.9i −1 ,

(31)

i

for each = 1, 2, · · ·, m. Next, the temperature decreases according to this equation. Additionally, the proposed SA-based permutation algorithm uses the temperature as a control parameter to determine the probability of accepting a worse candidate permutation. t ) ≥ E(P t ), the candidate permutation is accepted as the If E(Pcan cur t ) < E(P t ), the candidate current permutation. Conversely, if E(Pcan cur permutation is accepted with the following probability:



Prob = exp

− E(P t ) Ti



.

(32)

This statement implies that small decreases in the optimal evaluation value are more likely to be accepted than large decreases. Furthermore, the probability of accepting a worse candidate permutation is larger in the beginning, but this probability decreases in later iterations as the temperature decreases. Specifically, when the current temperature Ti is high, the most moves to worse permutations (a smaller value for the optimal evaluation value) will be accepted. In contrast, when the current temperature Ti approaches zero, nearly all moves to worse permutations will be rejected. The SA-based permutation algorithm allows movements for worse permutations with a certain probability, especially in the hightemperature stage, and thus, this algorithm can avoid becoming trapped in a local optimum. It is worthwhile to mention the interrelationship among criteria in certain real-world cases. The proposed SA-based permutation method may more appropriately handle MCDA problems based on the assumption that criteria are independent. Nevertheless, some practical MCDA problems may be characterized by interdependent criteria that even exhibit feedback-like effects [68]. The

J.-C. Wang, T.-Y. Chen / Applied Soft Computing 36 (2015) 57–69

63

Fig. 2. Example of the neighborhood structure.

existence of interrelationships (i.e., interdependence and feedback) among criteria results in a possible limitation in the proposed method. Once an interrelationship is found among criteria in the practical MCDA problem, we must consider the influences of various criterion settings and employ different fuzzy membership functions. Furthermore, the signed-distance-based approach possibly requires modification to conduct concordance and discordance analyses and tackle interdependent and feedback issues in real-world situations. Accordingly, a valuable direction for future research is suggested that focuses on the modification of the SA-based permutation method to address the MCDA problems characterized by interdependent criteria and their feedback-like effects. 3.3. Proposed algorithm In the context of IT2 TrFNs, the SA-based permutation method for solving an MCDA problem under incomplete preference information can be summarized in the following series of steps: Step 1: Formulate an MCDA problem. Specify the alternative set Z = {z1 , z2 , . . ., zm } and the criterion set C = {c1 , c2 , . . ., cn }, which is divided into CI and CII . Step 2: Select appropriate linguistic variables or other data collection tools to establish the IT2 TrFN evaluative rating Aij in (15) for the alternative zi ∈ Z with respect to criterion cj ∈ C. Step 3: Ask the decision maker’s preference information in terms of weak order, strict order, difference order, interval bound, and ratio bound in (9)–(13), respectively, and construct the set  . Then, relax the constraints in  to  by introducing the deviation variables, as depicted in (22)–(26). Step 4: Compute the signed distance sd(Aij ) for all zi ∈ Z and cj ∈ C. Identify the concordance set CSϕ and the discordance set DSϕ using (19) and (20), respectively, for the pairwise partial rankings, where , ϕ = 1, 2, . . ., m, and  = / ϕ. Step 5: Set the ς value, and let the total number of iterations tmax = mς . The initial temperature Tmax is defined in (30). Let T1 = Tmax , and the temperature decreases according to (31). 1 chosen at Step 6: Let t = 1. Generate an initial permutation Pcur random among all of the m! permutations. Establish the inte1 . Solve for the optimal grated programming model in (28) for Pcur 1 ). weight vector to determine the optimal evaluation value E(Pcur 1 ), and P 1 is stored as the approximate solution Let E(PSA ) = E(Pcur cur PSA . Step 7: If the stop condition is false, do Steps 8–11. Step 8: Use the exchange operator to generate the candidate pert mutation Pcan from the neighborhood solutions of the current t . Establish the model in (28) for P t . Solve for the permutation Pcur can optimal weight vector to determine the optimal evaluation value t ). E(Pcan

(new)

t ) ≥ E(P t ), then replace P t with P t t Step 9: If E(Pcan ≡ cur cur can (i.e., Pcur t Pcan ); otherwise, compute E(P t ) and Prob using (29) and (32), t ) < E(P t ), then replace P t by P t respectively. If E(Pcan cur cur can with the probability Prob. (new) t ), then replace P t Step 10: If E(PSA ) < E(Pcur ≡ SA with Pcur (i.e., PSA t ), and let E(P ) = E(P t ). Pcur SA cur Step 11: Test for the stop condition: if t = tmax , then stop; otherwise, replace t with t + 1 (i.e., t(new) ≡ t + 1), and continue.

4. Analysis of computational experiments This section presents an experimental analysis to examine the effectiveness and efficiency of the proposed SA-based permutation method. Computational experiments with notably large amounts of simulation data are designed to test the solution approach and compare the results with the best optimal evaluation value among all of the permutations. 4.1. Design of computational experiments The solution to the proposed model in (28) for all m! permutations is an NP-hard problem. The time required for the calculation of the signed-distance-based weighted differences associated with the criterion-by-criterion consistency should be considered. Additionally, solving the proposed model for all m! permutations would be costly with respect to time. Consequently, the solution can be obtained such that the computational complexity of the permutation-based outranking method is O(m! × n). The number of permutations increases rapidly with an increasing number of alternatives, and thus, it follows that the computational complexity of the proposed method is evidently high for much larger m. In contrast, the approximate solution PSA with the highest optimal evaluation value E(PSA ) during the tmax iterations represents the SA-based optimal ranking order of the alternatives. Note that the original computational complexity O(m! × n) has been reduced to O(mς ) (ς is a constant), which represents a significant improvement in the computational efficiency. This paper further conducts an experimental analysis to examine the implementation efficiency of the SA-based permutation algorithm and validate the correctness of the approximate solutions compared with the optimal all-permutation-based results. Instead of few numerical or empirical cases, we employed comprehensive computational experiments to cover the whole problem. To prevent the research from applying to only a few specific instances, we designed numerous varied instances in the hope that most possible instances of the problem could be covered. Accordingly, based on the experimental results, it is possible to make a legitimate inference in comparing the results of the proposed method with those of the permutation-based outranking method. The influence of different combinations of numbers of alternatives and criteria (especially the number of alternatives) on the correctness of the

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measure the correctness of the approximate solutions. We denote E(PSA ) and E(PPermu ) as the optimal evaluation values for the approximate solution PSA (yielded by the SA-based algorithm) and the optimal all-permutation-based solution PPermu (yielded by the permutation-based outranking method), respectively. The error percentage (Err%) of the approximate solution PSA relative to PPermu is defined as follows: Err% =

E(PPermu ) − E(PSA ) E(PPermu )

× 100%.

(34)

This percentage can measure the correctness of the SA-based permutation results. Specifically, the lower the error percentage Err%, the closer the optimal evaluation value E(PSA ) is to the true value E(PPermu ), and the better is the quality of the approximate solution PSA . Moreover, it indicates that the proposed SA-based permutation method can determine totally correct results in the case of Err%=0. 4.2. Experimental results and discussion Fig. 3. Sample data of IT2 TrFNs in the computational experiments.

approximate solutions and the improvement of the CPU time is also investigated through the analysis results of computational experiments. We applied notably large amounts of simulation data and implemented complete calculations to reach a tenable and solid conclusion for facilitating practical applications. Computer simulations using MATLAB were used to generate a range of MCDA problems with random data. The process of randomly generating an IT2 TrFN is summarized in the Appendix, and sample data for IT2 TrFNs are shown in Fig. 3. Following Steps I-IV in the Appendix, the data of the evaluative rating Aij can be generated randomly for the sake of constructing a problem of multiple criteria evaluations. In the computational experiments, IT2 TrFN evaluative ratings were randomly generated to form MCDA problems with all possible combinations of 3, 4, · · ·, 10 alternatives and 3, 4, · · ·, 10 criteria. Accordingly, 64 different instances were examined in the experimental analysis. For each instance, we randomly produced 100 sets of simulation data. Therefore, 6400 sets of experimental cases were generated. For brevity, we designate the number of criteria as the number of incompletely known pieces of information for the criterion weights. Specifically, n preference conditions and their corresponding lower bounds, upper bounds, and/or ratios are generated randomly in terms of the five basic ranking forms (i.e., weak order, strict order, difference order, interval bound, and ratio bound) to establish the set  . Let ς = 4 in the SA-based permutation algorithm. Thus, tmax = m4 . The proposed SA-based permutation algorithm was solved using MATLAB and simultaneously run on eight x64-based PCs with an Intel Core i7-2600K (3.4 GHz) CPU, 16 G of RAM, and the Windows 7 operating system. Note that the experiments not only focused on the reduced time costs but also on the correctness of the SA-based permutation results compared with the optimal all-permutationbased results. First, let TimeSA and TimePermu denote the average CPU times for implementing the SA-based algorithm and the permutation-based outranking method, respectively. The improvement percentage (Time%) of the average CPU times for PSA relative to Ppermu is defined as follows: Time% =

TimePermu − TimeSA × 100%. TimePermu

(33)

Next, to realize the quality of the SA-based permutation results compared with the optimal all-permutation-based results (i.e., the true values acquired by the original permutation-based outranking method), we use the concept of error percentages to

The comparison results of the average CPU times TimePermu and TimeSA are presented in Table 1. Regardless of the number of criteria, both the permutation-based outranking method and the proposed SA-based permutation method completed the solution very quickly in case of limited alternatives. Nevertheless, TimeSA was significantly smaller than TimePermu as the number of criteria increased. More specifically, TimePermu was 10 or 11 times as long as TimeSA when m = 8. For example, as indicated in Table 1, TimePermu was 236.4048 s (i.e., approximately 4 min) in the case of m = 8 and n = 5, while TimeSA was only 22.3464 s in the case of the former and latter. As for the situation of m = 9, the permutation-based outranking method was 58–60 times the average CPU time of the proposed method under different criterion settings. Taking m = 9 and n = 7 as an example, the average CPU times to solve the permutation-based outranking method and the proposed method were 2503.5505 s (i.e., approximately 42 min) and 43.0202 s (less than 1 min), respectively. Note that the difference between TimePermu and TimeSA was drastically enlarged when m = 10, especially in the cases of n = 3, 9, and 10. For example, solving the case of m = 10 and n = 5 required 21,029.9758 s (i.e., approximately 5.8 h) and 52.6798 s (less than 1 min), on average, using the permutation-based outranking method and the proposed method, respectively. Moreover, an average of 26,042.3033 s (i.e., approximately 7.2 h) and 6.8223 s were spent implementing the two comparative methods in the case of m = 10 and n = 9, respectively. As shown in Table 1, the implementation efficiency of the proposed method is very high compared to the permutation-based outranking method, whereas the influence of n on the average CPU time is relatively unobvious except for the case of m = 10. The results of the mean improvement percentage (mean Time%) for each m × n combination are reported in the upper half of Table 2. Obviously, finding all-permutation-based solutions would result in drastically increased CPU times as the number of alternatives increases. In contrast, the SA-based permutation algorithm could overcome this difficulty because the mean Time% reached approximately 54.15–56.03% in the case of m = 7 and 90.12–99.97% in the cases of m = 8, 9, and 10, as shown in the upper half of Table 2. Fig. 4 depicts comparative results of the mean improvement percentages with respect to the average CPU times in the experimental analysis. As revealed in Fig. 4(a), the effect of the number of alternatives is evident in all of m × n combinations. However, the number of criteria has little effect on the mean improvement percentages. Fig. 4(b) shows rather consistent shapes with respect to the number of alternatives. Specifically, the mean improvement percentages increased significantly as the m value increased.

J.-C. Wang, T.-Y. Chen / Applied Soft Computing 36 (2015) 57–69

65

Table 1 Comparison analysis of the average CPU time (unit: second). m

n 3

4

5

6

7

8

Results of the average CPU time (TimePermu ) for implementing the permutation-based outranking method 0.0319 0.0364 0.0350 0.0395 0.0419 3 4 0.1299 0.1439 0.1381 0.1428 0.1504 0.6923 0.6432 0.7140 0.7016 0.7553 5 6 4.0237 4.3533 4.3625 4.0981 4.4356 29.4596 28.8938 30.6630 31.3651 31.7592 7 218.2590 209.3053 236.4048 259.3343 257.3065 8 2414.1718 2067.6493 2166.0037 2309.4357 2503.5505 9 10 18,257.1233 19,584.2853 21,029.9758 21,990.6617 22,168.2324 Results of the average CPU time (TimeSA ) for implementing the SA-based permutation method 3 0.4279 0.4723 0.4573 0.5094 4 1.3248 1.4848 1.3969 1.4670 3.4498 3.1635 3.4915 3.4637 5 6.7178 7.6736 7.6546 7.1506 6 13.5059 12.9356 13.4830 13.8467 7 21.1597 20.1633 22.3464 24.4649 8 41.4641 35.0050 35.9418 39.7606 9 4.8274 48.5640 52.6798 57.3666 10

Fig. 4(c) compares the mean improvement percentages with respect to various numbers of criteria that appear as similar shapes. These shapes have a commonality, i.e., the mean improvement percentages remain unchanged as the number of criteria increases. In general, for large m values, the mean improvement percentages are much larger than those for the other curves. In contrast, the number of criteria has an insignificant effect on the mean improvement percentages. The results of the mean error percentage (mean Err%) for each m × n combination are presented in the lower half of Table 2. The mean Err% is equal to 0.00% in all of the cases of m = 3, 4, 5, and 6. The mean Err% reached approximately 0.00–0.17% in the cases of m = 7 and 8 and 0.13–1.95% in the cases of m = 9 and 10. As shown in the lower half of Table 2, the SA-based permutation algorithm yields the optimal all-permutation-based results for the decisionmaking problems with small values of m. Although the mean error percentages slightly increase for much larger m, the approximate solutions yielded by the proposed algorithm are notably close to the optimal all-permutation-based solutions for most of the cases. This finding supports the correctness of the approximate solutions and demonstrates the high quality of the SA-based permutation results.

0.5471 1.5619 3.7671 7.4974 14.3055 24.7338 43.0202 58.0564

9

10

0.0387 0.1662 0.8123 4.7514 33.6077 285.5764 2004.6664 24,149.6414

0.0446 0.1743 0.8402 5.2295 35.3435 273.3320 2523.9033 26,042.3033

0.0380 0.1706 0.9239 4.5386 35.2085 306.7107 2441.0924 27,590.3625

0.5021 1.6715 4.0554 8.1793 14.9318 27.2619 34.6033 63.0818

0.5767 1.8045 4.2264 9.0766 15.9023 26.0216 42.9633 6.8223

0.4978 1.7651 4.6655 8.0487 15.9274 30.3057 41.3044 7.2170

The comparative results of the mean error percentages in the experimental analysis are shown in Fig. 5. Fig. 5(a) presents the change in the mean error percentages in all of m × n combinations. Generally, the curves in Fig. 5(b) have common shapes and trends with respect to the number of alternatives. Specifically, the mean error percentages increased as the number of alternatives increased when m ≥ 7. As depicted in Fig. 5(c), when m = 8, 9, and 10, the shapes for the mean error percentages were somewhat irregular with respect to the number of criteria. Additionally, the mean error percentage in the case of m = 10 was the largest. In this experimental analysis, we set the total number of iterations tmax = m4 , which is much smaller than the number of permutations for much larger m. Thus, the mean error percentages in the cases of large m values were obviously larger than those in the cases of small m values. If we want to improve the accuracy rate of the SAbased solutions, we can employ a larger ς value to increase the iterative computations; nevertheless, this approach would require much CPU time. In general, the mean error percentages were much smaller in most of the m × n combinations. Additionally, the number of alternatives has a significant effect with respect to the mean error percentages, but the number of criteria does not.

Table 2 Summarized experimental results. unit: %. m

n 3

Results of the mean Time% −1239.40 3 4 −919.56 −398.33 5 −66.96 6 54.15 7 90.31 8 9 98.28 99.97 10 Results of the mean Err% 3 0.00 0.00 4 0.00 5 6 0.00 0.01 7 0.09 8 0.15 9 0.51 10

4

5

6

7

8

9

10

−1196.05 −932.09 −391.85 −76.27 55.23 90.37 98.31 99.75

−1205.74 −911.87 −389.02 −75.46 56.03 90.55 98.34 99.75

−1189.56 −927.28 −393.72 −74.48 55.85 90.57 98.28 99.74

−1205.25 −938.81 −398.76 −69.03 54.96 90.39 98.28 99.74

−1197.41 −905.53 −399.24 −72.15 55.57 90.45 98.27 99.74

−1191.66 −935.60 −403.05 −73.56 55.01 90.48 98.30 99.97

−1209.08 −934.33 −404.99 −77.34 54.76 90.12 98.31 99.97

0.00 0.00 0.00 0.00 0.04 0.12 0.20 0.45

0.00 0.00 0.00 0.00 0.00 0.00 0.13 1.11

0.00 0.00 0.00 0.00 0.00 0.01 0.13 1.07

0.00 0.00 0.00 0.00 0.00 0.14 0.30 1.84

0.00 0.00 0.00 0.00 0.03 0.07 0.51 1.91

0.00 0.00 0.00 0.00 0.04 0.17 0.28 1.31

0.00 0.00 0.00 0.00 0.00 0.03 0.40 1.95

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J.-C. Wang, T.-Y. Chen / Applied Soft Computing 36 (2015) 57–69

Fig. 4. Results of the mean improvement percentages of the average CPU times.

Next, more comparative discussions with other relevant approaches are provided to illustrate the advantages of the proposed SA-based permutation method. As mentioned previously, Chen and Wang [12] proposed an interval-valued fuzzy permutation method to address MCDA problems. They made use of interval-valued fuzzy sets to address imprecise decision information, and all of the computations were performed on ordinary intervals (not interval-valued membership functions) and not on IT2 FSs. Therefore, their method does not address data based on interval type-2 fuzzy numbers. At present, the MCDA approaches developed by Chen et al. [29] and Wang et al. [30] belong to the permutation-based outranking methodology in the context of IT2 FSs. Specifically, Chen et al. [29] developed an extended QUALIFLEX method to address medical decision-making problems within the IT2 TrFN

Fig. 5. Results of the mean error percentages.

environment. They used successive permutations of all possible rankings of alternatives and recognized several indices for all permutation rankings of the alternatives. Based on a signed distance-based approach, they established the concepts of concordance/discordance indices, weighted concordance/discordance indices, and comprehensive concordance/discordance indices as evaluative criteria of the chosen hypothesis for ranking the alternatives. On the other hand, Wang et al. [30] developed a likelihood-based QUALIFLEX method based on IT2 TrFNs for addressing MCDA problems. They introduced an extended concept of likelihoods of interval type-2 fuzzy preference relations. To evaluate the permutations of alternatives corresponding to

J.-C. Wang, T.-Y. Chen / Applied Soft Computing 36 (2015) 57–69

criterion-by-criterion consistency, they used a likelihood-based comparison approach to determine the concordance index and the comprehensive concordance index. The best permutation and its corresponding optimal ranking order of the alternatives can be acquired through a signed distance-based method or an optimal membership-degree-determination method. Although the extended QUALIFLEX method [29] and the likelihood-based QUALIFLEX method [30] can manage data based on IT2 TrFNs, they require complicated computations and have a tedious procedure to identify relevant concordance/discordance indices. In contrast, the proposed SA-based permutation method does not require determining any concordance/discordance index. We adopt a more simple and effective approach to acquiring the sets of concordance and discordance using the inequality relations between the signed distances of corresponding evaluative ratings. Additionally, Chen and Wang [12], Chen et al. [29], and Wang et al. [30] did not discuss the issue that the information on the criterion weights is only partially known. Incomplete preference information is realistic in numerous real-world situations because of time pressure, lack of knowledge or data, intangible or nonmonetary criteria, limited attention and information-processing capabilities of the decision makers, and complex and uncertain environments [60]. In contrast to the methods developed by Chen and Wang [12], Chen et al. [29], and Wang et al. [30], the proposed method provides a more flexible approach capable of manipulating incompletely known or even conflicting information with respect to the criterion importance. Therefore, compared to other established methods, the proposed method makes more contributions to real-world applications in the field of the permutation-based outranking methodology. Finally, the most distinct advantage of the proposed method over the existing methods is high implementation efficiency. As stated previously, the main drawback of the permutation-based outranking methodology is that the solution for all possible permutations is an NP-hard problem. It is a vast and time-consuming process when we implement the interval-valued fuzzy permutation method [12], the extended QUALIFLEX method [29], or the likelihood-based QUALIFLEX method [30] in the case of many alternatives. On the contrary, the developed method, which has demonstrated effectiveness and efficiency in applications via comprehensive experimental analysis, can determine a polynomial time solution for the total completion time problem. Therefore, the proposed method is recommended for solving MCDA problems in the IT2 FS context. In short, comparative discussions have verified the effectiveness and efficiency of the SA-based permutation method within the IT2 TrFN environment. In particular, the difficulty of implementing the computations for much larger m can be significantly overcome with the help of the proposed algorithm. 4.3. Additional discussion This subsection provides additional discussion on the proposed method. First, the IT2 TrFN data in the computational experiments were designed and randomly generated to construct a range of MCDA problems. To provide more contributions for practical implications, how to obtain IT2 TrFN data in the real world is discussed in this subsection. In general, human decision-making behavior is always subjective to a certain extent. Because individuals make decisions and perform actions according to what they perceive as reality, it is important to take human subjectivity into account as part of the decision-making process [29]. By means of the linguistic rating system, the analyst can conveniently investigate the decision maker’s opinions about the performance (i.e., evaluative ratings) of alternatives with respect to various criteria. Numerous studies

67

have presented useful linguistic rating systems for converting decision makers’ linguistic responses into proper IT2 TrFNs [69], such as five-point linguistic scales [27], seven-point linguistic scales [28,53], and nine-point linguistic scales [29,55]. The linguistic data can be converted into IT2 TrFNs using these existing transformation standards for the sake of addressing imprecision and uncertainty issues. Consider the linguistic rating system introduced by Zhang and Zhang [28] as an example. Their rating system is expressed on a seven-point scale anchored by “very low” and “very high.” Moreover, their linguistic rating system contains the corresponding IT2 TrFNs necessary to measure the alternative ratings. If a decision maker expresses the performance of an alternative with respect to a criterion using a linguistic term “medium high,” the corresponding IT2 TrFN is designated as [(0.55, 0.65, 0.65, 0.75; 0.8), (0.5, 0.6, 0.7, 0.8; 1)] according to Zhang and Zhang’s proposed transformation standards. IT2 TrFNs can efficiently express linguistic evaluations or assessments by objectively transforming them into numerical variables [28,69]. Therefore, it is suggested that linguistic variables be used to describe the evaluative ratings of alternatives and convert them into IT2 TrFNs for practical applications. The next issue is the applicable scope of the problem size in practical applications. As indicated in the experimental results, the proposed SA-based permutation method and its meta-heuristic algorithm can not only significantly improve the implementation efficiency but also produce a polynomial time solution for the total completion time problem. Thus, compared to the established permutation-based methods, the current method can be applicable to most real-life MCDA problems. Consider a consumer decision-making problem as an example. There may be literally hundreds of different brands or different variations (e.g., models) of the same brand. However, making a selection from a sample of all possible brands or variations is a human characteristic that helps simplify the decision-making process [70]. An evoked set is those brands or variations a consumer will evaluate for the solution and according to which he or she will make a purchase decision within a particular product category [71]. In fact, many consumers do not know much about the technical aspects of products, and they have only a few major brands in memory. A cross-national study found that people generally include only a few products in their consideration, although this amount varies by product category and country [72]. Solomon [73] also indicated that consumers often consider a small number of alternatives, especially with all the choices available to them. Thus, regardless of the total number of brands (or variations) in a product category, the evoked set often consists of the small number of brands (or variations) the consumer is familiar with, remembers, and finds acceptable. Accordingly, the proposed method can be definitely applied to address practical consumer decision-making problems although there are hundreds of different brands (or variations) in a particular product category. 5. Conclusions In the context of an IT2 TrFN framework, this paper proposed an SA-based permutation method for solving MCDA problems under incomplete preference information. This paper conducted an analysis of computational experiments to validate the correctness of the approximate solution compared with the optimal all-permutationbased result. As shown in the experimental results, the proposed SA-based permutation algorithm can indeed determine a satisfactory solution for the MCDA problems with large values of m in a short period of time. In addition to SA, selected heuristic and meta-heuristic approaches (e.g., tabu search, genetic algorithms, evolution programming, evolution strategies, genetic programming, particle swarm optimization, ant colony optimization, etc.)

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can be employed to implement the proposed method. For example, Diabat [74] developed a hybrid genetic/SA algorithm to address the nonlinear problem. Future research can focus on the hybrid algorithm for solving the integrated nonlinear programming model for all m! permutations. More noteworthy is how to improve the performance of each criterion and dimension. In the current paper, the purpose of the proposed SA-based permutation method is to manage MCDA problems involving multiple criteria evaluations and ranking/selection of alternatives, not improving the alternatives. The proposed method can distinguish a relatively good solution from the existing alternatives in numerous practical MCDA problems. Consider consumer decision making as an example. Many consumers can scarcely improve the performance or correct the features of the existing alternatives in a specific product or service category due to lack of expertise, technical skills, or bargaining power with the vendor/manufacturer. Thus, the proposed method can still make a practical contribution to such MCDA problems within an uncertain and imprecise environment. However, the emphasis in the MCDA field has gradually shifted from ranking and selection of alternatives to performance improvement. Thus, future studies can discuss the applicability of extending the SA-based permutation method to investigate how to improve the alternatives. Additionally, Zavadskas and Turskis [9] and Liou and Tzeng [10] reviewed numerous MCDA methods and presented several novel concepts and trends in the MCDA field. These studies also afford perspectives on valuable future research. Thus, it is anticipated that the proposed method can be further improved in the future by incorporating these new concepts, including the use of aspiration levels, non-additive/super-additive models, and a systematic approach to problem solving [10]. Acknowledgements The authors acknowledge the assistance of the respected editor and the anonymous referees for their insightful and constructive comments, which helped to improve the overall quality of the paper. The authors are grateful for grant funding support from the Taiwan Ministry of Science and Technology (MOST 102-2410-H182-013-MY3) and Chang Gung Memorial Hospital (BMRP 574) during the study completion. Appendix A. Appendix The process of randomly generating an IT2TrFN is summarized in the following steps: Step I: Generate five real numbers, 1U , 2U , · · ·, and 5U , which are uniformly distributed over the interval [0,1]. Let U =



U =1 

for

of Aij is designated as follows:  = 1, 2, . . ., 5. The argument aU ij

aU = ij

U 5U

for  = 1, 2, 3, 4.

(A.1)

Step II: Generate five real numbers, 1L , 2L , . . ., and 5L , which are uniformly distributed over the interval [0,1]. Let L =



L =1 

for

 = 1, 2, . . ., 5. The argument aLij of Aij is designated as follows:

aLij =

L 5L

(aU − aU ) + aU 4ij 1ij 1ij

for  = 1, 2, 3, 4.

Step III: Generate the height hLA . ij

(A.2)

Step III-1: Let the real numbers, 1 , 2 , and min , be defined as follows:



1 = min

aL2ij − aU 1ij aU − aU 2ij 1ij

 2 = min

aL3ij − aU 4ij aU − aU 3ij 4ij





,1

,

(A.3)

,

(A.4)

 ,1



min = min 1 , 2 .

(A.5)

Step III-2: Generate the height hLA that is uniformly distributed ij

over the interval [0, min ]. Step IV: Generate the height hU that is uniformly distributed over A the interval [max , 1], where



max



= max

hLA

ij

1

,

hLA

ij

2

ij



.

(A.6)

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